V. Bentkus

On the dependence of the Berry–Esseen bound on dimension

Abstract Let X be a random vector with values in R d . Assume that X has mean zero and identity covariance. Write β= E |X| 3 . Let Sn be a normalized sum of n independent copies of X. For Δ n = sup A∈ C | P {S n ∈A}−ν(A)| , where C is the class of convex subsets of R d , and ν is the standard d-dimensional normal distribution, we prove a Berry–Esseen bound Δ n ⩽400d 1/4 β/ n . Whether one can remove or replace the factor d1/4 by a better one (eventually by 1), remains an open question.

On Hoeffding’s inequalities

In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M n = X 1 + ... + X n of bounded independent random variables X j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P{M n ≥ x} ≤ cP{S n ≥ x}, where c is an absolute constant and S n = e 1 + ... + e n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ Ρ where the survival function x → P{S n ≥ x} has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P{S n ≥ x} special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere.

A Berry-Esséen bound for student's statistic in the non-I.I.D. case

We establish a Berry-Esséen bound for Students statistic for independent (nonidentically) distributed random variables. In particular, the bound implies a sharp estimate similar to the classical Berry-Esséen bound. In the i.i.d. case it yields sufficient conditions for the Central Limit Theorem for studentized sums. For non-i.i.d. random variables the bound shows that the Lindeberg condition is sufficient for the Central Limit Theorem for studentized sums.

The Accuracy of Gaussian Approximation in Banach Spaces

Let B be a real separable Banach space with norm || · || = || · || B . Suppose that X, X 1, X 2, … ∈ B are independent and identically distributed (i.i.d.) random elements (r.e.’s) taking values in B. Furthermore, assume that EX = 0 and that there exists a zero-mean Gaussian r.e. Y ∈ B such that the covariances of X and Y coincide.

A Lyapunov-type Bound in Rd

Let

On the asymptotical behavior of the constant in the Berry-Esseen inequality

X_1,\ldots,X_n

Limiting distributions of the non-central t-statistic and their applications to the power of t-tests under non-normality

be independent random vectors taking values in

On probabilities of large deviations in Banach spaces

{\bf R}^d

Optimal Error Estimates in Operator-Norm Approximations of Semigroups

such that

A New Method for Approximations in Probability and Operator Theories

{{\bf E} X_k =0}